2026 Theses Doctoral

Nonlinear Operator Equations in Hilbert Spaces: Foundations, Optimal Recovery, and Active Learning.

Lin, Daozhe

Nonlinear operator equations of the form 𝓕(𝑢) = 𝑣 are fundamental to the mathematical modeling of complex physical systems. Historically, research has focused on the solving the unknown state 𝑢 given a known forward operator 𝓕 and measurement 𝑣. However, with the rise of data-driven scientific computing, equal attention is now given to operator learning: approximating the unknown operator 𝓕 itself from a finite set of input-output data pairs (𝑢, 𝑣). Both paradigms face severe theoretical and computational bottlenecks, most notably the challenge of ensuring rigorous convergence from finite discrete measurements and overcoming the infinite-dimensional curse of dimensionality. This thesis provides a unified mathematical framework for both solving and learning nonlinear operators, grounded in the theory of optimal recovery, active learning, and Reproducing Kernel Hilbert Spaces (RKHS).

The thesis is structured around four primary contributions. First, we solve the nonlinear equation by developing a comprehensive framework for nonlinear optimal recovery in Hilbert spaces. This approach provides rigorous well-posedness, stability guarantees, and finite-dimensional tractability for recovering the state variable 𝑢 when the governing nonlinear operator 𝓕 is known.

Second, we transition this optimal recovery philosophy to the data-driven machine learning setting. We formulate operator learning from inexact, finite measurements as a regularized optimal recovery problem within a vector-valued RKHS. Crucially, we prove a Representer Theorem for operator learning, which guarantees the existence of an optimal operator and reduces the inherently infinite-dimensional optimization to a tractable, finite-dimensional computational task with explicit convergence rates.

Third, we address the practical bottleneck of operator learning: the infinite-dimensional curse of dimensionality that plagues standard passive sampling. Focusing on a broad class of mappings defined as operators of "neural network type,'' we propose a novel active learning framework. By utilizing adaptively chosen point queries to estimate local gradient covariance matrices, this strategy systematically uncovers the underlying functional active subspace. We provide theoretical guarantees demonstrating that this active querying effectively reduces the infinite-dimensional approximation problem to a bounded, finite-dimensional optimal recovery task.

Finally, to rigorously justify the capacities of the hypothesis classes and the theoretical bounds utilized throughout the preceding chapters, the thesis concludes by establishing the foundational information-theoretic limits of these function spaces. By rigorously analyzing the relationship between metric entropy (covering numbers) and embeddability within RKHS, we quantify the intrinsic spatial complexity. Together, these four works provide a rigorous, end-to-end mathematical foundation for optimally solving and learning complex nonlinear operator equations.